Apparatus and method for remote current sensing

ABSTRACT

An apparatus for remotely sensing the currents flowing in a set of substantially parallel conductors carrying N independent AC currents comprises N magnetic field sensors M 1 , M 2  positioned to take mutually independent measurements of the magnetic field generated by the conductors and provide corresponding signals. Processing means  12 ′ or  12 ″ derive further signals, respectively corresponding to the conductor currents, based upon the sensor signals and the location and angular orientation of each sensor relative to each conductor.

BACKGROUND OF THE INVENTION

This invention relates to an apparatus and method for remotely sensingthe alternating currants (AC) in a set of substantially parallelconductors from the magnetic fields generated by these currents in thevicinity of the conductors.

This invention relates to an apparatus and method for remotely sensingthe alternating currents (AC) in a set of substantially parallelconductors from the magnetic fields generated by these currents in thevicinity of the conductors.

SUMMARY OF THE INVENTION

According to the present invention there is provided an apparatus forremotely sensing the currents flowing in a se of substantially parallelconductors carrying N independent AC crrents, the apparatus comprising Nmagnetic field sensors positioned to take mutually independentmeasurements of the magnetic field generated by the conductors andprovide corresponding signals, and processing fews for deriving furthersignals, respectively corresponding to the conductor currents, basedupon the sensor signals and the location and angular orientation of eachsensor relative to each conductor.

Preferably, in the case where the further signals include harmoniccomponents of the fundamental frequency of the conductor currents, theprocessing means includes means for deriving the harmonic components ofthe further signals, adjusting the amplitudes of the harmonic componentsto reduce any distortion produced by said harmonic components in thesensors, and recombining the adjusted frequency components with thefundamental frequency components to produce said further signals withreduced distortion.

Preferably the means for deriving harmonic frequency components of thesensor signals comprises fourier analysis means.

By “positioned to take mutually independent measurements” we mean thatnone of the sensors provides data which is substantially the same as orsimply a linear combination of the data provided by the other(s). Inpractice this means that no two sensors have axes with the same angularorientation relative to the conductors and are in proximity when rightprojected onto a plane perpendicular to the conductors.

In this connection it is to be understood that the axis of a magneticsensor is that direction relative to the sensor which, when orientatedparallel to the lines of force of a fluctuating magnetic field passingthrough the sensor, would provide the maximum induced signal in thesensor for that magnetic field, and the plane of the sensor is a planepassing through the sensor normal to its axis.

Preferably the plane of each sensor is substantially parallel to theconductors.

The sensors are preferably coils, the plane of the coil being the planethrough the geometric centre of the coil parallel to the turns of thecoil and the axis of the coil being the direction through the coilcentre normal to such plane.

The invention further provides a method for remotely sensing thecurrents flowing in a set of substantially parallel conductors carryingN independent AC currents, the method comprising positioning N magneticfield sensors to take mutually independent measurements of the magneticfield generated by the conductors and provide corresponding signals, andderiving further signals, respectively corresponding to the conductorcurrents, based upon the sensor signals and the location and angularorientation of each sensor relative to each conductor.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the invention will now be described, by way of example,with reference to the accompanying drawings, in which;

FIG. 1 is a schematic view of an apparatus according to a firstembodiment of the invention for measuring the currents in an AC railtraction system with two overhead catenaries and return currents in therails;

FIG. 2 is a diagram illustrating the geometric parameters which aremeasured with respect to the apparatus of FIG. 1 and used in thecalculations according to Appendix 1;

FIG. 3 is a block diagram of the electronic circuitry within theinstrument of FIG. 1;

FIG. 4 shows how the processing of the signals from the sensing coils isdivided, in the embodiment, between the instrument and the PC;

FIG. 5 is a diagram illustrating the geometric parameters which aremeasured with respect to the remote sensing of the currents in the caseof an overhead power line;

FIG. 6 illustrates the use of Fourier analysis on the signals derivedfrom the sensing coils;

FIG. 7 is a schematic view of an apparatus according to a secondembodiment of the invention, including a post-mounted instrument and anassociated PC;

FIG. 8 is a diagram illustrating the geometric parameters which aremeasured and used in the calculations according to Appendix 2;

FIG. 9 is a block diagram of the electronic circuitry within theinstrument of FIG. 7; and

FIG. 10 shows how the processing of the signals from the sensing coilsis divided, in the second embodiment, between the instrument and the PC.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

Referring to FIGS. 1 to 6 of the drawings, an apparatus for measuringthe currents in the case of a two track AC rail traction system includesan instrument 10 and an associated PC 12′ or 12″. The instrument 10comprises an enclosure or housing 14 containing two magnetic fieldsensors M1 and M2. In this embodiment the sensors are coils. Theinstrument 10 is mounted on a catenary support (not shown) of the ACrail traction system or may be mounted on a post next to the AC railtraction system.

As seen in FIG. 2, in the present embodiment the traction systemcomprises two overhead substantially parallel catenaries 50-1, 50-2 eachassociated with a respective pair of parallel rails 52-1, 52-2. Eachcatenary together with its respective pair of rails constitutes acurrent loop with primary current in the catenary and return current inthe rails. Since the currents in the two catenaries 50-1, 50-2 areindependent, two sensing coils, i.e. the coils M1 and M2, positioned totake mutually independent measurements are necessary in the presentembodiment. However, the invention is applicable to parallel conductorAC systems having any number N of independent currents, in which casethe corresponding number N of magnetic field sensors, positioned to takemutually independent measurements, is used. This generalisation of theinvention is given in Appendix 1. Naturally the invention is not limitedto the sensing of currents in AC traction systems, and the applicationof the invention generally to overhead power lines and specifically to,for example, a 3 phase overhead power line with up to 4 wires(conductors), which requires three magnetic field sensors, is also givenin Appendix 1 with reference to FIG. 5.

Referring again to FIG. 2, the instrument 10 has a reference point P anda notional, but known, reference plane S passing through the point P.The instrument 10 is fixed such that the reference plane S issubstantially parallel to the catenaries 50-1, 50-2 and rails 52-1, 52-2(hereinafter all referred to as conductors). In practice the referenceplane S is approximately vertical, as shown, but this is not necessary.The coils M1 and M2 are arranged within the housing 14 such that theplane of each coil is also substantially parallel to the conductors.

Within the housing 14 the location of each coil M1 and M2 relative tothe reference point P is known (it is determined during manufacture).Each coil location is given by ΔX_(c) and ΔY_(c) (FIG. 2) which are theoffsets of the centre of the respective coil M1 or M2 from theinstrument reference point P, the ΔY value being measured parallel tothe instrument reference plane S and the ΔX value being measuredperpendicular to the instrument reference plane S. These are measuredpositive in the conventional sense, i.e. positive to the right and upfrom the reference point. The angular orientation θ_(c) of each coilrelative to the reference plane S is also known, θ_(c) being the anglebetween the plane of the coil M1 or M2 and the instrument referenceplane S. These angles are measured positive in the counter clockwisedirection from the reference plane.

In the general case where N independent currents are to be sensed the Nsensing coils are placed and oriented to obtain independent measures ofthe magnetic field in the vicinity of the conductors. To achieve thisthe coil orientation angles are varied typically through 90° to obtaindifferent magnetic field measures near the same location. Coils are thendisplaced to a sufficiently different location to obtain additionaldifferentiation between the readings. Such displacement would betailored to specific applications depending on the arrangement of theexternal conductors, the desired sensitivity and the desired compactnessof the instrument. For ease of data analysis the coils can be orientedwith their planes parallel and perpendicular to the instrument referenceplane, i.e. at orientation angles θ_(c) of 0° and 90°. However, they canbe oriented at any arbitrary orientation angle θ_(c) as desired providedtheir planes are substantially parallel to the conductors. The key pointis that all coils are accurately oriented, or their orientation isaccurately measured, and that their location is accurately known,preferably to a precision of at least 1 mm. In the present embodimentthe two coils M1 and M2 are orientated respectively parallel andperpendicular to the reference plane. The coils can be separated in thedirection parallel to the conductors to avoid mutual interference.

When the housing 14 is fixed in position as aforesaid, it is necessaryto know the respective radial distance R_(w) of each conductor from thereference point P, and also the respective angular displacement α_(w) ofeach conductor from the reference plane S. This is most preferablyachieved by providing the instrument 10 with a docking station in whichan ultrasonic measuring device can be installed to measure the radialdistances to the conductors and rotating this to measure the angles.These angles are measured positive in the counter clockwise directionfrom the reference plane.

When all these values are known, it will be appreciated that one has aprecise knowledge of the location and angular orientation of each coilM1, M2 relative to each conductor.

In use a respective instantaneous voltage V_(c) (t) will be induced ineach coil M1, M2 which is proportional to the instantaneous value of themagnetic field at the respective coil. In the case of coil, sensors theinduced voltage will also be proportional to the frequency of individualcomponents or harmonics in the magnetic field. The coils can be wound onferrite cores to improve sensitivity provided this does not cause mutualdistortion between the two coils if they are in close proximity.

The enclosure 14 contains electronics 28 powered by a battery 30, andhas a standard RS232 interface 32 for communication over a direct cableconnection 100 with the serial port of a PC 12″, for example a laptopPC, or over a communications link such as a modem or GSM module 102 to aPC 12′ via a modem 104 at the PC. It is to be understood that the PCs12′ and 12″ illustrate two different ways of connecting a PC to theinstrument 10, and in general only one such PC will be used at any giventime. The port 32 is programmable for a range of Baud rates(1,200-33,000 bps), and implements appropriate flow control and errorchecking.

The instrument 10 is capable of communicating with a host program 106(FIG. 4) in the PC 12′ or 12″ to download data to the PC on request. Itis also capable of receiving from the PC user selectable parameters suchas the integration time intervals, to be described, of having theinstrument clock checked or reset, of indicating the time span overwhich data has been logged, etc.

The host program 106 in turn is capable of communicating with theinstrument 10 for assessing the status of the instrument and its datacontent. It is capable of initiating a download of the data from theinstrument and of controlling the data flow. It is capable of receivingand processing geometric data (to be described) for the site inquestion. It is capable of converting the raw instrument data to currentvalues as necessary. It is capable of archiving all the data for a sitein a database entry for that site. It is capable of retrieving,displaying and graphing data for a site as required.

FIG. 3 is a block diagram of the electronics 28. The voltages from thecoils M1, M2 are first clamped in protection circuits 34 to provideover-voltage protection, to avoid damaging the instrument electronics inthe event of a disturbance. They are protected against the equivalent ofa fault current of 20,000 amps. The instrument does not have to measuresuch currents. It is only necessary to avoid being damaged by them.

The voltages from the coils are then filtered as appropriate by filters36 to pass the fundamental 50-60 Hz currents and all harmonics ofinterest. This normally includes harmonics up to the fortieth, i.e.2-2.4 kHz. A pass band of 30-2,500 Hz. is therefore appropriate.

Next the sensing coil voltages are captured in simultaneous sample andhold circuits 38. Sequential sampling would introduce a phase angleerror corresponding to the delay, which would affect the accuracy of theresults. A minimum sampling frequency of 5000 Hz is preferably used.

A block of data is sampled. This would typically be 5 cycles but can bemore or less down to 1 cycle depending on the specific application. Foraccuracy in subsequent arms calculations and fourier transformation ofthe data it is important to sample an integral number of fundamentalcycles. The sampling frequency should be adapted to the supply frequencyto achieve an integral number of samples and cycles in a block of datafor varying supply frequency. This may entail using a separate analogfilter to extract the fundamental 50-60 Hz component from one of thecoils and counting the zero crossings on this signal to measure thefrequency.

The sampled signals are then passed through a multiplexer 40 forconversion to digital form in an ADC converter 42 for further digitalprocessing. The cycle is then repeated. For continuous monitoring thereshould be two processes running in parallel. One continuously samplesblocks of coil data and transfers these to buffers. The other readsthese buffers and processes the data to derive engineering results, i.e.the current values.

The instantaneous digital values V_(c) (t) are processed digitally in aprocessor 44, firstly by scaling the voltages, using coil constantsF_(c), to give magnetic field intensities H_(c) (t)and then by a lineartransformation to cancel out the effect of the conductor geometryrelative to the coils, see block 60, FIG. 4. This is detailed inAppendix 1. The scaling factors incorporate externally definablecalibration factors to facilitate periodic re-calibration of theinstrument. This produces signals h_(w) (t) close to the individualconductor currents.

The linear transformation 60 requires certain geometric factors G⁻¹_(wc). As detailed in Appendix 1, these are derived from the externalgeometric parameters R_(w), α_(w) for the conductors and from theinternal geometric data ΔX_(c), ΔX_(c), θ_(c), etc., for the coils. Theexternal geometric parameters are input manually to the host program 106loaded on the PC 12′ or 12″. The internal geometric data is storedinternally for the instrument. The host program calculates the compositegeometric factors (72, FIG. 4) and uploads them to the instrument 10.

In the case of coils as magnetic sensors the voltages induced in thesensing coils M1, M2 are determined by frequency in addition to themagnitude of the conductor currents. The sensed voltages areproportional to the product of current magnitude and frequency. Thus theharmonic currents produce signal components amplified by their frequencyor harmonic order, e.g. the tenth harmonic produces a signal ten timesgreater than an equivalent fundamental current. The signals h_(w) (t)must now be adjusted to compensate for the distortion produced by thisamplification of harmonics to accurately and completely derive theoriginal currents.

The signals h_(w) (t) are subject to fourier analysis 62, as shown inmore detail in FIG. 6. The fast fourier transform (FFT) may be usedwhere the number of samples in a data block is a power of 2, i.e. equals2^(N) where N is an integer. The sampling frequency can be arranged toachieve this. For example for nominal 50 Hz currents and 5 cycle datablocks this can be achieved with a sampling frequency of 5,120 Hz or 512samples per block. The fourier transform isolates the individualfrequency components in the signals h_(w) (t). For example thefundamental components of the signals will typically occur as the fifthelement in the FFT transformed data, where we have 5 cycles per block ofdata. The harmonic components will reside at integral multiples of thislocation, e.g. position 10 for the second harmonic in the case of 5cycles per block of data, etc. The magnitudes of each frequencycomponent can then be divided by the harmonic order, i.e. f/f_(s) wheref_(s) is the supply frequency, to cancel the frequency amplificationeffect.

An inverse fourier transform is then applied to the adjusted data tore-create the original currents. Note that the FFT transformationproduces complex number values at each frequency to incorporate both themagnitude and phase angle of the signal at each frequency. The inverseFFT should incorporate both the adjusted magnitudes and the originalphase angles to accurately reproduce the conductor currents as asequence of values over time.

The FFT transformation and its inverse are standard mathematicaltechniques. They are described in digital signal processing (DSP) booksand are available as standard functions in most DSP software libraries.The digital signal processing to compensate for harmonic amplificationeffects can be implemented in a DSP chip or in a DSP component in acomposite controller housing a DSP element. It can also be implementedin software in an embedded PC environment or in a high powermicro-controller.

The resulting values I_(w) (t) describe the independent conductorcurrents as a sequence of instantaneous values over the period coveredby the sampling block. The FFT transformation as a by product alsoproduces the fundamental current magnitudes and phase angles for each ofthe conductors. For three phase currents in the case of a 3 phaseoverhead power line the fundamental frequency values can be converted(64, FIG. 4) to symmetrical component values, i.e. to describe thepositive, negative and zero sequence components of the conductorcurrents. This is done using the conventional phase to sequencetransformation as described in the summary of equations. The FFTtransformation also produces magnitudes and phase angles for theharmonic components of the conductor currents. Any or all of this datacan be used for further processing, e.g. for power quality analysis orevent recording, load recording or fault analysis, etc., as indicated byblocks 66 to 70 in FIG. 4. For example, the instantaneous current valuescan be stored in response to an event trigger to record the currentwaveshapes associated with a disturbance or event such as a fault, i.e.in disturbance/event recorder or oscillography applications. Thepositive and zero sequence currents can be monitored to indicate faultsin fault passage indicator applications, etc. Any of the data can alsobe time stamped and stored in the memory 46 for later retrieval andprocessing.

Alternatively the data can be condensed by averaging it over userdefined time intervals typically 10 minutes. In this case theinstantaneous values are aggregated to compute root mean square (rms)values initially for each block and then for the blocks within the timeperiod of interest. The resulting rms values for the current informationor parameters of interest are time stamped or structured in a definedtime sequence and stored in memory 46 for subsequent retrieval andanalysis. In such cases there should be sufficient memory to store atleast four weeks of data at ten minute intervals.

Upon request of the host program 106 via the cable connection 100 orcommunications link 102/104, the instrument 10 uploads the data storedin memory 46 to the PC 12′ or 12″ where it is stored at 74. It is thenavailable for further processing 76 and/or archived or displayed 78. Thehost program 106 may carry out one or more of the functions 66 to 70instead of the instrument 10.

In addition to performing calculations and analysis on the data from theinstrument 10, the host program also provides programmable softwarecalibration factors for the sensing coils, since it is necessary toperiodically re-calibrate the instrument.

Also, it is desirable to have a low battery voltage software flag in theinstrument 10 to signal the condition to the host program at the nextlinkup.

The instrument 10 is preferably rated to measure 0-1000 amp conductorcurrents and may be constructed to operate over user-selectableranges,.e.g. 0-500 and 0-1000 amps.

The instrument 10 is capable of processing the signals from the sensingcoils to achieve an instrument accuracy of 1% relative to the balancedline currents. The overall accuracy of results, taking account of theexternal geometric factors and processing in the host program is 5%.

As mentioned above, in practice the angles θ_(c) between the planes ofthe coils and the instrument reference plane S are set to 0° or 90°. Thereason for this is that in such case Cos θ_(c) and Sin θ_(c) becomeeither 0 or 1 in the attached equations so that the terms are thereforesimplified accordingly. This leads to considerably easier and fastercalculation. However, as mentioned before, the coils can have anyangular orientation relative to the reference plane S and the fullequations are therefore given.

Even so, the attached equations deal only with the case where the planeof each coil is parallel to the conductors. However, it is quitepossible to develop the equations further to cater for the planes of thecoils not necessarily being parallel to the wires, and the invention isintended to cover that possibility.

The foregoing has described a general instrument capable of measuringcurrents in a wide variety of situations, such as the AC rail tractionsystem or overhead power line described.

A second embodiment of the invention, now to be described, is designedexpressly for the purpose of remotely measuring the currents flowing inan AC overhead power line with up to four substantially parallel wires(and hence up to three independent currents).

Referring to FIGS. 7 to 10 of the drawings, the apparatus according tothe second embodiment of the invention includes an instrument 110 and anassociated PC 112′ or 112″. The instrument 110 comprises an enclosure orhousing 114 containing three magnetic field sensors a, b and c. In thisembodiment the sensors are coils. The instrument 110 is mounted on apost (not shown) below the an AC overhead power line having foursubstantially parallel wires R, Y, B and N whose currents are to bemeasured. The post on which the instrument is mounted may the same poleor pylon carrying the wires.

Although the apparatus is capable of measuring the wire currents in ACpower distribution systems using less than four overhead wires, thedrawings show the apparatus in use with a system using foursubstantially parallel overhead wires R, Y, B and N where the wires R,Y, B carry three current phases respectively and the wire N is a neutralwire. The wires can be in any configuration provided they aresubstantially parallel to each other. The wires are typically mountedhorizontally next to one another 1 m apart, and in such case theinstrument 110 is preferably mounted from 3 to 5 m, typically about 4 m,below them. It is not necessary for the instrument 110 to be locatedsymmetrically relative to the wires.

Referring to FIG. 8, the instrument 110 has a reference point P and anotional, but known, reference plane S passing through the point P. Theinstrument 110 is fixed such that the reference plane S is parallel tothe wires R, Y, B and N. In practice the reference plane S isapproximately vertical, as shown, but this is not necessary. The coilsa, b, c are arranged within the housing 114 such that the plane of eachcoil is also parallel to the wires.

Within the housing 114 the location of each coil relative to thereference point P is known (it is determined during manufacture). Thecoil locations are given by (ΔX_(a), ΔY_(a)), (ΔX_(b), ΔY_(b)) and(ΔX_(c), ΔY_(c)) which are the offsets of the centres of the coils a, band c respectively from the instrument reference point P, the ΔY valuesbeing measured parallel to the instrument reference plane S and the ΔXvalues being measured perpendicular to the instrument reference plane S.

The angular orientation of each coil relative to the reference plane Sis also known. θ_(a), θ_(b) and θ_(c) are the angles between the planesof the coils a, b and c respectively and the instrument reference planeS. In practice, as shown in FIG. 8 and for reasons of simplification tobe described later, the coils are arranged such that θ_(a)=0°, θ_(b)=90°and θ_(c)=0° but this is not necessary and the coils can be arranged atany orientation relative to the plane S provided their planes are stillsubstantially parallel to the wires. Preferably, but not necessarily,for convenience of packaging in a compact housing 114 the coils a and bare located in proximity and at right angles to each other, while thecoil c is located above or below the coils a, b. The coils can beseparated in the direction parallel to the wires to avoid mutualinterference.

When the housing 114 is fixed in position as aforesaid, it is necessaryto know the respective radial distance R₁, R₂, R₃ and R₄ of thereference point P from each wire R, Y, B and N, and also the respectiveangular displacement α₁, α₂, α₃ and α₄ of each wire R, Y, B and N fromthe reference plane S. This is most preferably achieved by providing theinstrument 110 with a docking station in which an ultrasonic measuringdevice can be installed to measure the radial distances to the wires androtating this to measure the angles.

Alternatively, the instrument can be installed on a pole with theinstrument aligned with the pole, i.e. its reference plane running alongthe axis of the pole. The height below the central wire is measured withsuch as an ultrasonic instrument. The lateral displacements of the wiresform the pole centre and any vertical displacements above or below thecentral wire are known from construction standards. This data can beentered in a PC host programme to derive the wire geometry relative tothe instrument.

When all these values are known, it will be appreciated that one has aprecise knowledge of the location and angular orientation of each coila, b and c relative to each wire R, Y, B and N.

In use a respective instantaneous voltage H_(a) (t), H_(b) (t) and H_(c)(t) will be induced in each coil a, b and c which is proportional to theinstantaneous value of the magnetic field at the respective coil. Thecoils can be wound on ferrite cores to improve sensitivity provided thisdoes not cause mutual distortion between the two coils a, b if they arein close proximity.

The enclosure 114 contains electronics 128 powered by a battery 130, andhas a standard RS232 interface 132 for communication over a direct cableconnection 200 with the serial port of a PC 112″, for example a laptopPC, or over a communications link such as a modem or GSM module 202 to aPC 112′ via a modem 204 at the PC. It is to be understood that the PCs112′ and 112″ illustrate two different ways of connecting a PC to theinstrument 110, and in general only one such PC will be used at anygiven time. The port 132 is programmable for a range of Baud rates(1,200-19,200 bps), and implements appropriate flow control and errorchecking.

The instrument 110 is capable of communicating with a host program 206(FIG. 10) in the PC 112′ or 112″ to download data to the PC on request.It is also capable of receiving from the PC user selectable parameterssuch as the integration time intervals to be described, of having theinstrument clock checked or reset, of indicating the time span overwhich data has been logged, etc.

The host program 206 in turn is capable of communicating with theinstrument 110 for assessing the status of the instrument and its datacontent. It is capable of initiating a download of the data from theinstrument and of controlling the data flow. It is capable of receivingand processing geometric data (to be described) for the site inquestion. It is capable of converting the raw instrument data to currentvalues as necessary. It is capable of archiving all the data for a sitein a database entry for that site. It is capable of retrieving,displaying and graphing data for a site as required.

FIG. 9 is a block diagram of the electronics 128. The voltages from thecoils a, b, c are first clamped in protection circuits 134 to provideover-voltage protection, to avoid damaging the instrument electronics inthe event of a disturbance. They are protected against the equivalent ofa symmetrical fault current of 20,000 amps in the overhead line. Theinstrument does not have to measure such currents. It is only necessaryto avoid being damaged by them.

The voltages from the coils are then filtered by low pass filters 136 topass fundamental, i.e. 50 to 60 Hz data, and to cut off harmonics abovethis frequency range.

Next the sensing coil voltages are captured in simultaneous sample andhold circuits 138. Sequential sampling would introduce a phase angleerror corresponding to the delay, which would affect the accuracy of theresults. A minimum sampling frequency of 500 Hz is used with a preferredsampling frequency of 1,000 Hz. The sampled signals are then passedthrough a multiplexer 140 for conversion to digital form in an ADCconverter 142.

From each set of three instantaneous digital coil values H_(a) (t),H_(b) (t) and H_(c) (t) a processor 144 derives a respective set of sixinstantaneous derivative digital values h₁ (t), h₂ (t), h₃ (t), h₁₂ (t),h₁₃ (t) and h₂₃ (t), step 300, FIG. 10. The formulae for deriving thederivative values from the instantaneous digital coil values are givenin the attached Appendix 2.

It will be noted in Appendix 2 that the calculation of the derivativevalues requires certain geometric factors Δ_(1a), Δ_(1b), Δ_(1c),Δ_(2a), Δ_(2b), Δ_(2c), Δ_(3a), Δ_(3b) and Δ_(3c). These are derivedfrom the geometric parameters R₁, R₂, R₃, R₄, α₁, α₂, α₃ and α₄previously referred to. These geometric parameters are input manually tothe host program 206 (FIG. 10) loaded on the PC 112′ or 112″. The hostprogram calculates the geometric parameters and uploads them to theinstrument 110.

The derivative digital values so calculated are integrated by theprocessor 144 to derive equivalent rms values over user selected timeintervals, step 302, FIG. 10. The time interval options are 5, 10, 15,20 and 30 minutes, which are programmable into the instrument 110 fromthe host program 206 at set up time. The rms data is time stamped andlogged into memory 146, or structured in a defined time sequence inmemory, for later retrieval, step 304. There is sufficient memory tostore four weeks of data at ten minute intervals. In the event ofrunning out of memory space the instrument overwrites the oldest datastored.

Upon request of the host program 206 via the cable connection 200 orcommunications link 202/204, the instrument 110 uploads the rms valuesin memory 146 to the PC 112′ or 112″. As seen in FIG. 10, step 306,using the rms values the host program 206 calculates three so-calledsequence currents, a positive sequence current I₊which is in effect theaverage current in the wires, a negative sequence current I⁻which is ineffect the load unbalance current, and a zero-sequence current I₀ whichis in effect the leakage current to earth in the event of a fault or theneutral current in the case of a four wire circuit. From these sequencecurrents the host program 206 calculates, in amperes, the actualcurrents I_(R), I_(Y), I_(B) and I_(N) in the wires, step 308. Theformulae which the host program uses to calculate these are set out inAppendix 2.

The current values thus calculated may be displayed on the PC monitor bythe host program 206, or stored in PC memory, step 310.

In addition to calculating the currents in the wires, the host programalso provides programmable software calibration factors for the sensingcoils, since it is necessary to periodically re-calibrate theinstruments.

Also, it is desirable to have a low battery voltage software flag in theinstrument 110 to signal the condition to the host program at the nextlinkup.

The instrument 110 is preferably rated to measure 0-1000 amps as abalanced current in all three wires of an overhead line or as a faultcurrent in any one wire with return through the earth, and may have twoor more user selectable ranges, e.g. 0-500 and 0-1000 amps. The usercould select between them at set up time.

The instrument 110 is capable of processing the signals from the sensingcoils to achieve an instrument accuracy of 1% relative to the balancedline currents. The overall accuracy of results, taking account of theexternal geometric factors and processing in the host program is 5%.

Although the foregoing has described all three coils a to c as beinglocated below the wires R, Y, B and N, this is only for ease ofaccommodating them in a single relatively small enclosure 114. Each ofthe coils could be located above or below the wires, provided that theymeet the positional requirements specified above.

Although the second embodiment has been described in use with the mostdemanding case of four overhead wires R, Y, B and N, it can also be usedto measure the currents in two- or three-wire power distributionsystems. In the two-wire system, which uses wires R and N or two of thephases for example, the geometric parameters R₃ and R₄ are set toinfinity, while in the three-wire system, which uses wires R, Y and B,the geometric parameter R₄ is set to infinity.

As mentioned above, in practice the angles θ_(a), θ_(b) and θ_(c)between the planes of the coils a, b and c respectively and theinstrument reference plane S are set such that θ_(a)=0_(°, θ) _(b)=90°and θ_(c)=0°. The reason for this is that in such case Cos θ_(a), Sinθ_(a), etc. become either 0 or 1 in the attached equations so that theCos α_(a1), etc. terms are therefore simplified accordingly. This leadsto considerably easier and faster calculation. However, as mentionedbefore, the coils can have any angular orientation relative to thereference plane S and the full equations are therefore given.

Even so, the attached equations deal only with the case where the planeof each coil is parallel to the wires. However, it is quite possible todevelop the equations further to cater for the planes of the coils notnecessarily being parallel to the wires, and the invention is intendedto cover that possibility.

The enclosure 14 or 114 in which the instrument 10 or 110 is housed is arobust enclosure for field use. With access panels closed it providesprotection to IP64 in line with IEC standard 529. The enclosure andseals are capable of withstanding continuous exposure to sunlight, rain,frost, etc., without undue degradation over a life of 10 years.

The mounting arrangement for the instrument 10 or 110 can take the formof a bracket that could be attached to a wooden pole with coach screws.Alternatively lugs and mounting bands could be employed to attach theinstrument to any type of pole. It should be as easy to remove theinstrument as to install it. The mounting arrangement should hold theinstrument in a secure position on the pole so that there is no movementor tilting in the event of winds.

In addition to the external commas port 32 or 132 the instrument has anon/off switch. The on/off switch may also serve as a reset switch tocope with the possibility of a lock up during instrument set up. Thecomms port and on/off switch may be located behind such as a perspexdoor. It should be possible to apply a standard wire seal to the door toavoid tampering with the instrument.

The instrument preferably has two indicating LEDs, one to indicate poweron and the other to indicate that the instrument is fully set up andlogging data. These can also be installed behind a transparent door.

The instrument 10 or 110 is powered by the battery 30 or 130 to supportcontinuous normal operation over a period of five years. The battery isa standard readily available size. It should be possible to replace thebattery in the field with standard tools. The instrument can be removedfrom the pole to fit a new battery.

The instrument 10 or 110 should preferably operate accurately within thefollowing service conditions:

Maximum temperature: 40 deg C

Minimum temperature: −10 deg C

Relative humidity: 0-85%

Elevation above sea level: 0-1,000 m.

Maximum wind gust velocity: 50 m/s.

The invention is not limited to an apparatus where the processing issplit between an on-site instrument and a remote processor. The entireapparatus could be accommodated in a single housing on-site. Thedivision of data processing and functionality between an instrument andhost software in a PC can vary with specific applications. Also thebasic current measurement functionality could be implemented as adiscrete entity and used as a front end in or with other conventionalinstrumentation to feed remotely sensed current data into the instrumentfor further analysis or processing, e.g. power quality analysis, faultindication, etc.

Also, the apparatus need not be fixed in position on a pole or the like.Embodiments in the form of a hand-held device are possible. In that casethe geometric parameters would be measured by a measuring device, e.g.ultrasonic, laser, etc., built into the instrument, readings being takensimultaneously with the readings of the magnetic field.

The invention is not limited to the embodiment described herein whichmay be modified or varied without departing from the scope of theinvention.

APPENDIX 1 Geometric Factors:

Given:

A set of conductors (wires) carrying N independent currents. Each ofthese currents flows out in a primary conductor denoted by the subscriptW for wire. Each of the currents can return in one or more returnconductors denoted by W′ for the first such return conductor and W″,etc. for any subsequent conductors.

R_(w), α_(w) measure the conductor geometry of one of the primaryconductors relative to the instrument. These values respectively are theradial distances R_(w) from the instrument reference point to the centreof the conductor and the angular displacements α_(w) of the conductorfrom the instrument reference plane. The angular displacements aremeasured positive in the counter clockwise direction from the instrumentreference plane. R_(w′), α_(w′), R_(w″), α_(w″), etc. measure theconductor geometry of any associated return conductors relative to theinstrument. This gives us a set of conductor geometries as follows:

R₁, α₁, R₂, α₂, R₃, α₃, . . . R_(N), α_(N).

R_(1′), α_(1′), R_(2′), α_(2′), R_(3′), α_(3′), . . . R_(N′), α_(N′).

R_(1″), α_(1″), R_(2″), α_(2″), R_(3″), α_(3″), . . . R_(N″), α_(N″).

In addition for any return conductor(s) we define parameter(s) λ_(w′),λ_(w″), etc., which specify the fraction of the current in question thatreturns in each of these conductors. For example with two rails forreturn current we would typically have λ_(w′)=λ_(w″)=0.50.

A set of N magnetic field sensors or sensing coils. These are denoted bythe subscript C for coil ΔX_(c), ΔY_(c) are the coil offsets from theinstrument reference point to the geometric centres of these sensingcoils. These are measured positive from the reference point in theconventional sense, i.e. to the right and up. θ_(c) are the orientationsof the planes of these coils with the instrument reference plane. Theseare measured positive in the counter clockwise direction from theinstrument reference plane, e.g. a coil with its plane parallel to thereference plane has θ_(c)=0° or 180° depending on polarity. This givesus a set of coil geometries as follows:

ΔX₁, ΔY₁, θ₁, ΔX₂, ΔY₂, θ₂, ΔX₃, ΔY₃, θ₃, . . . ΔX_(N), ΔY_(N), θ_(N).

Transform the measured conductor geometry to each of the sensing coilsas follows:

R_(cw)=[R_(w) ²+ΔX_(c) ²÷ΔY_(c) ²+2. R_(w). (Sin α_(w). ΔX_(c)−Cosα_(w). ΔY_(c))].

Cos α_(cw)=[(R_(w). Cos α_(w)−ΔY_(c)). Cosθ_(c)+(R_(w). Sinα_(w)+ΔX_(c)). Sin θ_(c)]/ R_(cw).

C=1 . . . N and W=1 . . . N.

This produces a separate set of geometric data adapted to theorientations and positions of each of the sensing coils. Thus for Nsensing coils the complete external conductor geometry is transformed Ntimes.

Calculate the geometric terms:

G_(cw)=Cos (α_(cw))/ 2 πR_(cw)−λ_(w′). Cos (α_(cw′))/ 2 πR_(cw′)−λ_(w″).Cos (α_(cw″))/ 2 πR_(cw″)−. . .

C=1 . . . N and W=1 . . . N.

This produces a matrix of N ×N terms corresponding to the N sensingcoils and the N independent currents in the set of conductors.

Invert this N ×N matrix in the above terms to derive the geometricfactors G⁻¹ _(wc).

Note that in the particular case of a 3 phase 4 conductor circuit (FIG.5) we only have 3 independent currents. The current in the fourthconductor is dependent on that in the three phase conductors. Wetherefore only need 3 sensing coils to fully characterise the currentsin all the conductors.

In this case we do not have a separate return conductor for each of thephase currents. Most of the current, i.e. all the positive and negativesequence current, sums to zero in the load. The fourth conductor servesas a common return conductor for the residual or zero sequence currentfor all three phases. We formulate the geometric equations somewhatdifferently in this case. Let λ be the fraction of residual or zerosequence current returning in the fourth or neutral conductor. This istaken as 1.0 by default.

The external conductor geometry R_(w), α_(w) is measured relative to theinstrument for all 4 conductors. This produces 4 pairs of values. Thisdata is transformed to each of the 3 sensing coils as before to give:

R_(cw)=[R_(w) ²+ΔX_(c) ²+ΔY_(c) ²+2. R_(w). (Sin α_(w).ΔX_(c)−Cos α_(w).ΔY_(c))]

Cos α_(cw)=[(R_(w). Cos α_(w)−ΔY_(c)). Cos θ_(c)+(R_(w). Sinα_(w)÷ΔX_(c)). Sin θ_(c)]/ R_(cw).

C=1 . . . 3 and W=1 . . . 4.

Note also that in this case in practice we would take θ₁=0°, θ₂=90° andθ₃=0° so that Cos θ_(c), Sin θ_(c) above become either 0 or 1. Thissimplifies the expression for Cos α_(cw).

This produces 12 pairs of values. The geometric terms are alsocalculated as before:

G_(cw)=Cos (α_(cw))/ 2 πR_(cw).

C=1 . . . 3 and W=1 . . . 4.

This gives us a matrix with 3 rows and 4 columns. We use the fact thatthe current in the fourth conductor is the residual current from thethree phase conductors, i.e. the currents in all four conductors sum to0. Allowing for the fact that some of the residual current in practicemay be returning through the earth we have:

λ(I₁+I₂+I₃) +I₄=0.

We now add the coefficients of this euation to the geometric matrix,i.e. G_(4w)=[λ, λ, λ, 1]. We invert the full 4×4 matrix to get G⁻¹_(w4). Drop the rightmost column in this 4×4 matrix since the term thatwould be multiplied by these factors is 0. This gives a 4×3 matrix G⁻¹wcfor deriving each of the 4 conductor currents from the 3 coil readings.

In each case we now have the geometric factors G⁻¹ _(wc) for eliminatingthe effect of the conductor geometry from the sensing coil readings. Theresulting values are directly related to the conductor currents.

Remove Geometry:

Firstly apply the scaling factors for the coils to convert the signals,e.g. in volts, to units of magnetic field intensity, i.e. A/m. Thesescaling factors or constants F_(c) accommodate the numbers of turns ineach coil, calibration constants, etc.

Thus H_(c) (t)=F_(c). V_(c) (t).

Using the scaled sensing coil readings H_(c) (t) transform thesereadings to eliminate the conductor geometry:${h_{w}\quad (t)} = {{\sum\limits_{C = 1}^{N}\quad {G_{WC}^{- 1} \times H_{C}\quad (t)}} = {{G_{W1}^{- 1} \times H_{1}\quad (t)} + {G_{W2}^{- 1} \times H_{2}\quad (t)} + \ldots + {G_{WN}^{- 1} \times H_{N}\quad (t)}}}$W = 1  …  N.

Where fundamental frequency (50/60 Hz) quantities only are beingprocessed these values h_(w) (t) would be the actual conductor currentsI_(w) (t). Thus I_(w)(t)=h_(w) (t).

Fourier Analysis:

In advanced applications the desired currents can also have harmonic ortransient components. The instrument is monitoring the voltages inducedin the coils. These are proportional to frequency in addition to thecurrent magnitude. So the signal components associated with currentcomponents above or below the fundamental frequency are amplified inproportion to the ratio of their frequency to the fundamental frequency,i.e. by f/f_(s) where f_(s) is the fundamental frequency. We have tocompensate for the distortion produced by this effect.

We use fourier analysis to separate the frequency components of thecurrent signals. The amplitudes are divided by the ratio f/f_(s) tocancel the harmonic amplification. The adjusted values can then berecombined with an inverse fourier transform to produce the fullycompensated signals representing the true conductor currents I_(w). Thisprocess is illustrated schematically in FIG. 6.

Note that the above process can be performed for each of the currentindividually. However for the phase sequence conversion to derivesymmetrical component current values in the case of three phase currentsthe data for the three phases in question must be available to theconversion process.

The number of samples in a block of data is selected to be an integralpower of 2 so that the fast fourier transform (FFT) and its inverse canbe used. This reduces the computational effort in processing the data.

Process Data

The harmonic amplification adjustment process produces a set ofinstantaneous values describing the individual conductor currents I_(w)(t). This effectively described the wave shapes of the conductorcurrents to the precision allowed by the sampling rate. Thisinstantaneous data can be directly stored, displayed or made availableto other processes within the instrument or to other instruments forfurther analysis.

If desired rms values can be computed over specified integration periodsas follows:$I_{w}^{rms} = {\sqrt{\left\lbrack {\sum\limits_{t = 1}^{M}\quad {I_{w}^{2}\quad {(t)/M}}} \right\rbrack}.}$

This data would typically be time stamped and stored in memory.

The fourier analysis as a by-product extracts the harmonic currents andtheir phase angles over the period of a sampling block of data. Thisdata can likewise be stored, displayed or made available to otherprocesses within the instrument or to other instruments for furtheranalysis. Rms values can be computed over specified integration periods.This data can also be time stamped and stored in memory.

Sequence Currents

The fourier analysis as a by-product also extracts the fundamentalcurrents I_(F) and their phase angles θ_(F) over the period of asampling block of data. For three phase currents this data can beconverted to sequence or symmetrical component currents for the datablock in question. The phase current magnitudes I_(φ) and their phaseangles θ_(φ), where φ is a sub-script corresponding to the phases 1, 2,3 or R, Y, B, are most conveniently combined to form complex numberrepresentations of the phase currents in the form X+iY. Taking

α=−{fraction (1/2)}+i 3/2 and α₂=−{fraction (1/2)}−i 3/2 then

I₊=(I_(R)+α. I_(Y) +α².I_(B) )/3

I⁻=(I_(R)+α². I_(Y) +α. I_(B) )/3

I_(0 =(I) _(R)+I_(Y) +I_(B) )/3

where I₊is the positive sequence current representing balanced loadconditions, I⁻is the negative sequence current representing loadunbalance within the three phases and I₀ is the zero sequence currentrepresenting residual current either as a natural current from loadunbalance or fault current from one or more of the phases or acombination of both.

These symmetrical component currents can also be integrated or averagedover user designated time intervals to provide rms values as a profileover time. This data can also be time stamped and stored.

APPENDIX 2 Geometric Factors:

Given:

λ= Fraction of zero sequence current returning in the neutral conductor=1.0 by default.

(R₁, α₁), (R₂, α₂), (R₃ α₃) and (R₄, α₄) measure the wire geometryrelative to the instrument.

R₁, R₂, R₃ and R₄ are the radial distances from the instrument referencepoint P to the wires R, Y, B and N respectively, and α₁, α₂, α₃ and α₄are the angular displacements of the wires R, Y, B and N respectivelyfrom the instrument reference plane S measured positive in ananticlockwise direction.

(ΔX_(a), ΔY_(a)), (ΔX_(b), ΔY_(b)) and (ΔX_(c), ΔY_(c)) are the offsetsof the centres of the coils a, b and c respectively from the instrumentreference point P, the ΔY values being measured parallel to theinstrument reference plane S and the ΔX values being measuredperpendicular to the instrument reference plane S. These are measuredpositive in the conventional sense, i.e. positive to the right and upfrom the reference point P.

θ_(a), θ_(b) and θ_(c) are the angles between the planes of the coils a,b and c respectively and the instrument reference plane S measuredpositive in an anticlockwise direction. Note that in practice we takeθ_(a) =0°, θ_(b) =90° and θ_(c) =0° so that Cos θ_(a), Sin θ_(a), etc.become either 0 or 1. The Cos α_(a1), etc. terms are thereforesimplified accordingly.

Transform the measured wire geometry to each of the coil positions:

R_(a1)=[R₁ ² +ΔX_(a) ² +ΔY_(a) ² +2. R₁. (Sin α₁. ΔX_(a)−Cos α₁.ΔY_(a))]

R_(a2)=[R₂ ² +ΔX_(a) ² +ΔY_(a) ² +2. R₂. (Sin α₂. ΔX_(a)−Cos α₂.ΔY_(a))]

R_(a3)=[R₃ ² +ΔX_(a) ² +ΔY_(a) ² +2. R₃. (Sin α₃. ΔX_(a)−Cos α₃.ΔY_(a))]

R_(a4)=[R₄ ² +ΔX_(a) ² +ΔY_(a) ² +2. R₄. (Sin α₄. ΔX_(a)−Cos α₄.ΔY_(a))]

Cos α_(a1)=(R₁. Cos α₁−ΔY_(a)) / R_(a1). Cos θ_(a)+(R₁. Sin α₁+ΔX_(a)) /R_(a1). Sin θ_(a).

Cos α_(a2)=(R₂. Cos α₂−ΔY_(a)) / R_(a2). Cos θ_(a)+(R₂. Sin α₂+ΔX_(a)) /R_(a2). Sin θ_(a).

Cos α_(a3)=(R₃. Cos α₃−ΔY_(a)) / R_(a3). Cos θ_(a)+(R₃. Sin α₃+αX_(a)) /R_(a3). Sin θ_(a).

Cos α_(a4)=(R₄. Cos α₄=ΔY_(a)) / R_(a4). Cos θ_(a)+(R₄. Sin α₄+αX_(a)) /R_(a4). Sin θ_(a).

R_(b1)=[R₁ ²+ΔX_(b) ²+ΔY_(b) ²+2. R₁. (Sin α₁. ΔX_(b)−Cos α₁. ΔY_(b))]

R_(b2)=[R₂ ²+ΔX_(b) ²+ΔY_(b) ²+2. R₂. (Sin α₂. ΔX_(b)=Cos α₂. ΔY_(b))]

R_(b3)=[R₃ ²+ΔX_(b) ²+ΔY_(b) ²+2. R₃. (Sin α₃. ΔX_(b)=Cos α₃. αY_(b))]

R_(b4)=[R₄ ²+ΔX_(b) ²+ΔY_(b) ²+2. R₄. (Sin α₄. ΔX_(b)=Cos α₄. ΔY_(b))]

Cos α_(b1)=(R₁. Cos α₁−ΔY_(b)) / R_(b1). Cos θ_(b)+(R₁. Sin α₁+ΔX_(b)) /R_(b1). Sin θ_(b).

Cos α_(b2)=(R₂. Cos α₂−ΔY_(b)) / R_(b2). Cos θ_(b)+(R₂. Sin α₂+ΔX_(b)) /R_(b2). Sin θ_(b).

Cos α_(b3)=(R₃. Cos Δ₃−ΔY_(b)) / R_(b3). Cos θ_(b)+(R₃. Sin α₃+ΔX_(b)) /R_(b3). Sin θ_(b).

Cos α_(b4)=(R₄. Cos Δ₄−ΔY_(b)) / R_(b4). Cos θ_(b)+(R₄. Sin α₄+ΔX_(b)) /R_(b4). Sin θ_(b).

R_(c1)=[R₁ ²+ΔX_(c) ²+ΔY_(c) ²+2. R₁. (Sin α₁. ΔX_(c)−Cos α₁. ΔY_(c))]

R_(c2)=[R₂ ²+ΔX_(c) ²+ΔY_(c) ²+2. R₂. (Sin α₂. ΔX_(c)−Cos α₂. ΔY_(c))]

R_(c3)=[R₃ ²+ΔX_(c) ²+ΔY_(c) ²+2. R₃. (Sin α₃. ΔX_(c)−Cos α₃. ΔY_(c))]

R_(c4)=[R₄ ²+ΔX_(c) ²+ΔY_(c) ²+2. R₄. (Sin α₄. ΔX_(c)−Cos α₄. ΔY_(c))]

Cos α_(c1)=(R₁. Cos α₁−ΔY_(c)) / R_(c1). Cos θ_(c)+(R₁. Sin α₁+ΔX_(c)) /R_(c1). Sin θ_(c).

Cos α_(c2)=(R₂. Cos α₂−ΔY_(c)) / R_(c2). Cos θ_(c)+(R₂. Sin α₂+ΔX_(c)) /R_(c2). Sin θ_(c).

Cos α_(c3)=(R₃. Cos α₃−ΔY_(c)) / R_(c3). Cos θ_(c)+(R₃. Sin α₃+ΔX_(c)) /R_(c3). Sin θ_(c).

Cos α_(c4)=(R₄. Cos α₄−ΔY_(c)) / R_(c4). Cos θ_(c)+(R₄. Sin α₄+ΔX_(c)) /R_(c4). Sin θ_(c).

Calculate the geometric terms:

A={Cos (α_(a1))/ 2 πR_(a1)−{fraction (1/2.)}Cos (α_(a2))/ 2πR_(a2)−{fraction (1/2.)}Cos (α_(a3))/ 2 πR_(a3)}

B={−3/2.Cos (α_(a2))/ 2 πR_(a2)+3/2.Cos (α_(a3))/ 2 πR_(a3)}

C={Cos (α_(a1))/ 2 πR_(a1)+Cos (α_(a2))/ 2 πR_(a2)+Cos (α_(a3))/ 2πR_(a3)−3 λCos (α_(a4))/ 2 πR_(a4).}

D={Cos (α_(b1))/ 2 πR_(b1)−{fraction (1/2.)}Cos (α_(b2))/ 2πR_(b2)−{fraction (1/2.)}Cos (α_(b3))/ 2 πR_(b3)}

E={−3/2.Cos (α_(b2))/ 2 πR_(b2)+3/2.Cos (α_(b3))/ 2 πR_(b3)}

F={Cos (α_(b1))/ 2 πR_(b1)+Cos (α_(b2))/ 2 πR_(b2)+Cos (α_(b3))/ 2πR_(b3)−3 λCos (α_(b4))/ 2 πR_(b4).}

G={Cos (α_(c1))/ 2 πR_(c1)−{fraction (1/2.)}Cos (α_(c2))/ 2πR_(c2)−{fraction (1/2.)}Cos (α_(c3))/ 2 πR_(c3)}

H={−3/2.Cos (α_(c2))/ 2 πR_(c2)+3/2.Cos (α_(c3))/ 2 πR_(c3)}

I={Cos (α_(c1))/ 2 πR_(c1)+Cos (α_(c2))/ 2 πR_(c2)+Cos (α_(c3))/ 2πR_(c3)−3 λCos (α_(c4))/ 2 πR_(c4).}

Invert the 3×3 matrix in the above terms to derive the geometricfactors:

Δ=A. (E.I−F.H)+B. (F.G−D.I)+C. (D.H−E.G)

Δ_(1a)=(E.I−F.H) / Δ

Δ_(1b)=(H.C−B.I) / Δ

Δ_(1c)=(B.F−C.E) / Δ

Δ_(2a)=(F.G−D.I) / Δ

Δ_(2b)=(A.I−C.G) / Δ

Δ_(2c)=(C.D−A.F) / Δ

Δ_(3a)=(D.H−E.G) / Δ

Δ_(3b)=(B.G−A.H) / Δ

Δ_(3c)=(A.E−B.D) / Δ

These geometric calculations are only performed at instrument set up orif the instrument is moved, e.g. in the case of a mobile unit.

Instantaneous Values:

Using coil readings H_(a)(t), H_(b)(t) and H_(c)(t) transform thereadings to eliminate the geometry:

h₁(t)=Δ_(1a) . H_(a)(t)+Δ_(1b) . H_(b)(t)+Δ_(1c) . H_(c)(t)

h₂(t)=Δ_(2a) . H_(a)(t)+Δ_(2b) . H_(b)(t)+Δ_(2c) . H_(c)(t)

h₃(t)=Δ_(3a) . H_(a)(t)+Δ_(3b) . H_(b)(t)+Δ_(3c) . H_(c)(t)

h₁₂(t)=h₁(t) . h₂(t)

h₁₃(t)=h₁(t) . h₃(t)

h₂₃(t)=h₂(t) . h₃(t)

Derive rms values for: H₁, H₂, h₃, h₁₂, H₂₃ and h₂₃, i.e.

h₁=[(Σ₁ ^(N)h₁ ²(t)0/ N], etc.

Sequence Currents:

Using h₁, H₂, h₃, h₁₂, h₂₃ and h₂₃ derive the sequence values:

χ=(h₁ ²+H₂ ²) / 2

δ=[(h₁ ²−h₂ ²)₂ / 4+h₁₂ ²]

I₊₌½{[χ+δ]+[χ−δ]}

I⁻⁼½{[χ+δ]−[χ−δ]}

I0=h₃

Sin θ⁻=h₁₂ / (2. I_(+.)I⁻)

Cos θ_(.)=(h₁ ²) / (4 . I_(+.) I⁻)

ε=h₁₃ / (h₁ . h₃)

γ=h₂₃ / (h₂ . h₃)

W=(I₊+I_(−.) Cos θ⁻) / h₁

X=I_(31 .)Sin (θ⁻) / h₁

Y=(I₊−I_(−.)Cos θ⁻) / h₂

Z=−I_(−.)Sin (θ⁻) / h₂

Sin (θ₀)=(Z. ε+W . γ) / (W . Y +X . Z)

Cos (θ₀)=(Y. ε−X . γ) / (W . Y +X . Z)

Wire Currents:

Using I₊, I⁻, I₀, Sin θ_(−, Cos θ) _(−, Sin (θ) ₀) and Cos (θ₀) derivethe individual wire currents:

I_(R)=[(I₊+L Cos (θ⁻)+I₀ Cos (θ₀))²+(I⁻Sin (θ⁻)+I₀ Sin (θ₀))²]

I_(Y)=[(I₊+L (−½Cos (θ⁻)+3/2. Sin (θ⁻))+I₀ (−½Cos (θ₀)−3/2. Sin(θ₀)))²+(I⁻(−½Sin (θ⁻) −3/2. Cos (θ⁻))+I₀ (−½Sin (θ₀)+3/2. Cos (θ₀)))²]

I_(B)=[(I₊+I⁻(−½Cos (θ⁻)−3/2. Sin (θ⁻))+I₀ (−½Cos (θ₀)+3/2. Sin(θ₀)))²+(I⁻(−½Sin (θ⁻)+3/2. Cos (θ⁻))+I₀ (−½Sin (θ₀)−3/2. Cos (θ₀)))²]

I_(N)=3 λI₀

What is claimed is:
 1. An apparatus for remotely sensing the currentsflowing in a set of substantially parallel conductors carrying Nindependent AC currents, the apparatus comprising a portable housingcontaining N magnetic field senors each positioned at a known locationand orientation in the housing to take mutually independent measurementsof the magnetic field generated by the conductors and providecorresponding sensor signals, and processing means for deriving furthersignals respectively corresponding to the conductor currents, theprocessing means receiving inputs defining location and orientation ofthe housing relative to each conductor, and the further signals beingderived by the processing means based upon the sensor signals and thelocation and angular orientation of each sensor relative to eachconductor as determined by the known location and orientation of eachsensor in the housing and the received inputs.
 2. An apparatus asclaimed in claim 1, wherein the processing means includes means forderiving the harmonic components of the further signals, adjusting theamplitudes of the harmonic components to reduce any distortion producedby said harmonic components in the sensors, and recombining the adjustedfrequency components with the fundamental frequency components toproduce said further signals with reduced distortion.
 3. An apparatus asclaimed in claim 2, wherein the means for deriving harmonic frequencycomponents of the sensor signals comprises Fourier analysis means.
 4. Anapparatus as claimed in claim 3 wherein said apparatus is adapted toremotely sense the currents flowing in an overhead power line comprisingfour wires carrying three independent AC currents, the apparatus havingthree magnetic field sensors.
 5. An apparatus as claimed in claim 4wherein the axes of two of the sensors are substantially parallel to oneanother and the axis of the third sensor is substantially perpendicularto the axes of the first two sensors.
 6. An apparatus as claimed inclaim 3 wherein the magnetic field sensors are coils.
 7. An apparatus asclaimed in claim 2 wherein said apparatus is adapted to remotely sensethe currents flowing in an overhead power line comprising four wirescarrying three independent AC currents, the apparatus having threemagnetic field sensors.
 8. An apparatus as claimed in claim 7 whereinthe axes of two of the sensors are substantially parallel to one anotherand the axis of the third sensor is substantially perpendicular to theaxes of the first two sensors.
 9. An apparatus as claimed in claim 2wherein the magnetic field sensors are coils.
 10. An apparatus asclaimed in claim 1 wherein said apparatus is adapted to remotely sensethe currents flowing in an overhead power line comprising four wirescarrying three independent AC currents, the apparatus having threemagnetic field sensors.
 11. An apparatus as claimed in claim 10, whereinthe axes of two of the sensors are substantially parallel to one anotherand the axes of the third sensor is substantially perpendicular to theaxes of the first two sensors.
 12. An apparatus as claimed in claim 10wherein the magnetic field sensors are coils.
 13. An apparatus asclaimed in claim 1, wherein the magnetite field sensors are coils. 14.An apparatus as claimed in claim 1 wherein the axes of two of thesensors are substantially parallel to one another and the axis of thethird sensor is substantially perpendicular to the axes of the first twosensors.
 15. A method for remotely seizing the currents flowing in a setof substantially parallel conductors carrying N independent AC currents,the method comprising mounting a portable housing in a fixed positionrelative to the conductors, the housing containing N magnetic fieldsensors each positioned at a known location and orientation in thehousing to take mutually independent measurements of the magnetic fieldgenerated by the conductors and provide corresponding signals, measuringlocation and orientation of the housing relative to each conductor,providing the measured location and orientation of the housing relativeto each conductor as inputs to a processing means, and using theprocessing means to derive and further signals, respectivelycorresponding to the conductor currents, based upon the sensor signalsand the location and angular orientation of each sensor relative to eachconductor as determined by the known location and orientation of eachsensor in the housing and the provided inputs.